Link to actual problem [10840] \[ \boxed {y^{\prime \prime }+y^{\prime } a +b \left (-b \,x^{2 n}-a \,x^{n}+x^{n -1} n \right ) y=0} \]
type detected by program
{"unknown"}
type detected by Maple
[[_2nd_order, _with_linear_symmetries]]
Maple symgen result This shows Maple’s found \(\xi ,\eta \) and the corresponding canonical coordinates \(R,S\)\begin{align*} \\ \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= {\mathrm e}^{-\frac {a x n +b \,x^{n +1}+x a}{n +1}}\right ] \\ \left [R &= x, S \left (R \right ) &= {\mathrm e}^{\frac {x \left (b \,x^{n}+a \left (n +1\right )\right )}{n +1}} y\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \left (\int {\mathrm e}^{\frac {a x n +2 b \,x^{n +1}+x a}{n +1}}d x \right ) {\mathrm e}^{-\frac {a x n +b \,x^{n +1}+x a}{n +1}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {x \left (b \,x^{n}+a \left (n +1\right )\right )}{n +1}} y}{\int {\mathrm e}^{\frac {x \left (2 b \,x^{n}+a \left (n +1\right )\right )}{n +1}}d x}\right ] \\ \end{align*}